Inner and Outer Regularity of a Finite Borel Measure on a Metric Space, and Lipschitz Approximation of Indicators

Let $(X,d)$ be a metric space with $X$ nonempty, let $\mathcal{T}_{d}$ be its topology of open subsets, and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra; by claim 1 of Borel Measurability and Bounded Integration on a Metric Space every open subset of $(X,d)$ and every closed subset of $(X,\mathcal{T}_{d})$ belongs to $\mathcal{B}(X)$. Let $\mu$ be a Borel measure on $(X,d)$ with $\mu(X)<\infty$, and regard $\mathbb{R}$ as a metric space with the absolute-value metric, its Borel $\sigma$-algebra being written $\mathcal{B}(\mathbb{R})$. For $B\in\mathcal{B}(X)$ let $\mathbf{1}_{B}:X\to\mathbb{R}$ be the indicator of $B$.

1. (Approximation from inside and from outside)(i) Let $B\in\mathcal{B}(X)$ and let $\varepsilon$ be a positive real number. Then there are a set $F$ closed in $(X,\mathcal{T}_{d})$ and a set $U$ open in $(X,d)$ with $F\subseteq B\subseteq U$ and

$$\mu(U\setminus F)\le\varepsilon .$$

2. (Inner regularity by closed sets)(ii) Let $B\in\mathcal{B}(X)$. Then the set of real numbers $\{\mu(F):F\subseteq B\text{ and }F\text{ is closed in }(X,\mathcal{T}_{d})\}$ is nonempty and bounded above, and $\mu(B)$ is its least upper bound.

3. (Outer regularity by open sets)(iii) Let $B\in\mathcal{B}(X)$. Then the set of real numbers $\{\mu(U):B\subseteq U\text{ and }U\text{ is open in }(X,d)\}$ is nonempty and bounded below, and $\mu(B)$ is its greatest lower bound.

4. (Lipschitz approximation of an indicator)(iv) Let $B\in\mathcal{B}(X)$ and let $\varepsilon$ be a positive real number. Then there are a map $h:X\to\mathbb{R}$ that is Lipschitz with a constant and satisfies $0\le h(x)\le1$ for every $x\in X$, and a set $N\in\mathcal{B}(X)$ with $\mu(N)\le\varepsilon$, such that

$$h(x)=\mathbf{1}_{B}(x)\qquad\text{for every }x\in X\setminus N .$$